Minimum unrooted binary spanning tree

Given a graph $$G$$ with $$n$$ tip vertices, $$n-2$$ internal vertices and a cost on each edge $$C(v)$$, find a minimum spanning tree subject to degree constraints:

1. tips have degree $$1$$
2. internal vertices have degree $$3$$.

There are no edges between pairs of tips in the given graph. These constraints ensure an unrooted binary tree forms.

I thought to use the this algorithm, which I based off Prim's algorithm:

For n-2 iterations:
Find the pair of edges i and j neighbouring a single internal vertex that minimises the cost C(i)+C(j)
Add edges i and j to the tree